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Nijenhuis operators on 2D pre-Lie algebras and 3D associative algebras

Xiaoguang Zou, Xiang Gao, Chuangchuang Kang, Jiafeng Lü

TL;DR

This work provides a complete classification of Nijenhuis operators on all 2‑dimensional complex pre‑Lie algebras and 3‑dimensional complex associative algebras, organized into commutative/noncommutative families. It establishes a systematic pipeline: a Nijenhuis operator on a pre‑Lie algebra A induces a weight‑zero Rota‑Baxter operator on the sub‑adjacent Lie algebra g(A), which in turn yields skew‑symmetric CYBE solutions on the semidirect product g(A) ⋉_{ad^*} g(A)^*. The authors supply explicit, parameterized lists of operators for all relevant low‑dimensional algebras and demonstrate the CYBE construction with a complete worked example on a 2‑D pre‑Lie algebra B1. The results offer concrete tools for generating CYBE solutions from finite‑dimensional algebraic data and point to scalable approaches for higher dimensions and related structures.

Abstract

In this paper, we describe all Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras. As an application, using these operators, we obtain solutions of the classical Yang-Baxter equation on the corresponding sub-adjacent Lie algebras.

Nijenhuis operators on 2D pre-Lie algebras and 3D associative algebras

TL;DR

This work provides a complete classification of Nijenhuis operators on all 2‑dimensional complex pre‑Lie algebras and 3‑dimensional complex associative algebras, organized into commutative/noncommutative families. It establishes a systematic pipeline: a Nijenhuis operator on a pre‑Lie algebra A induces a weight‑zero Rota‑Baxter operator on the sub‑adjacent Lie algebra g(A), which in turn yields skew‑symmetric CYBE solutions on the semidirect product g(A) ⋉_{ad^*} g(A)^*. The authors supply explicit, parameterized lists of operators for all relevant low‑dimensional algebras and demonstrate the CYBE construction with a complete worked example on a 2‑D pre‑Lie algebra B1. The results offer concrete tools for generating CYBE solutions from finite‑dimensional algebraic data and point to scalable approaches for higher dimensions and related structures.

Abstract

In this paper, we describe all Nijenhuis operators on 2-dimensional complex pre-Lie algebras and 3-dimensional complex associative algebras. As an application, using these operators, we obtain solutions of the classical Yang-Baxter equation on the corresponding sub-adjacent Lie algebras.
Paper Structure (19 sections, 24 theorems, 78 equations, 6 tables)

This paper contains 19 sections, 24 theorems, 78 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Nijenhuis algebras and Nijenhuis torsion
  3. Nijenhuis operators on pre-Lie algebras
  4. Outline of the paper
  5. Preliminaries
  6. Nijenhuis operators on 2-dimensional pre-Lie algebras
  7. Nijenhuis operators on 2-dimensional commutative pre-Lie algebras
  8. Nijenhuis operators on 2-dimensional non-commutative pre-Lie algebras
  9. Nijenhuis operators on 3-dimensional associative algebras
  10. Nijenhuis operators on 3-dimensional commutative associative algebras
  11. Nijenhuis operators on 3-dimensional non-commutative associative algebras
  12. Applications of Nijenhuis Operators to the CYBE
  13. From Nijenhuis operators to Rota-Baxter operators and CYBE solutions
  14. A complete worked example: from a 2D pre-Lie algebra $(B_1,\cdot)$ to an explicit $r$-matrix
  15. Sub-adjacent Lie algebras of pre-Lie algebras and associative algebras
  16. ...and 4 more sections

Key Result

Proposition 2.2

(BurdeBai-Meng) Let $\left(A, \cdot\right)$ be a 2-dimensional commutative pre-Lie algebra and $\left\{e_{1}, e_{2}\right\}$ be a basis of $A$. Then $(A,\cdot)$ isomorphic to one of the following algebras: $(A_1)$$e_1\cdot e_1=e_1,~e_2\cdot e_2=e_2$; $(A_2)$$e_1\cdot e_1=e_1,~e_1.e_2=e_2\cdot e_1=e_

Theorems & Definitions (44)

  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Proposition 2.7
  • proof
  • Theorem 3.1
  • proof
  • ...and 34 more